A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.

1, 1, 1, 3, 3, 6, 9, 12, 19, 27, 39, 54, 79, 107, 150, 209, 282, 387, 525, 707, 949, 1272, 1688, 2244, 2968, 3902, 5125, 6712, 8752, 11383, 14780, 19109, 24671, 31768, 40791, 52280, 66860, 85296, 108621, 138054, 175085, 221676, 280161, 353414, 445098, 559661

Offset

0,4

Comments

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..50

nLab, Young Diagram.

The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Example

The a(8) = 19 generalized Young tableaux:
  8   1 7   2 6   3 5   1 2 5   1 3 4
.
  1   2   3   1 2   1 5   1 3   1 4   2 3   1 2   1 2 3
  7   6   5   5     2     4     3     3     2 3   2
.
  1   1   1 2
  2   3   2
  5   4   3

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sqfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], SquareFreeQ]}]];
ptnplane[n_]:=Union[Map[primeMS, Join@@Permutations/@sqfacs[n], {2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&&And@@(UnsameQ@@@DeleteCases[Transpose[PadRight[#]], 0, {2}])&]], {y, IntegerPartitions[n]}], {n, 10}]

Crossrefs

Cf. A000085, A000219, A003293, A114736, A117433, A138178, A299968.

Cf. A323430, A323436, A323437, A323439, A323450.

Sequence in context: A136289 A128012 A289143 * A280240 A058628 A035528

Adjacent sequences: A323448 A323449 A323450 * A323452 A323453 A323454

Keyword

nonn

Author

Gus Wiseman, Jan 16 2019

Extensions

a(21)-a(45) from Seiichi Manyama, Aug 19 2020

Status

approved